Fractional Differential Equations Non-local Multi-point Boundary Value Problems

Fractional operators have a long history, having been mentioned by Leibnitz in a letter to L' Hospital in 1695. Referring to the question of fractional differentiation. Early mathematicians who contributed to fractional differential operators include Li-ouville, Riemann, and Holmgrem. In the last few decades many authors pointed out that fractional calculus are very suitable for the description of memory and hereditary properties of various materials and processes, such effects are in fact neglected in classi-cal models. Nowadays, fractional differential equations are increasingly used to model problems in acoustics and thermal systems, rheology and modelling of materials and mechanical systems, signal processing and systems identification, control and robotics, and other areas of application. Furthermore, the study of the nature of the solutions is also important.Fractional calculus is a specialized integral and derivative of arbitrary order and the application of the mathematical nature of the field, is the traditional promotion of integer order calculus.This paper investigates existence of solutions for a nonlinear multi-point boundary value problem for fractional differential equations.derivative is the standard Riemann-Liouville fractional derivative.The present paper constructs an appropriate space and employs the nonlinear functional analysis methods such as the cone theory, fixed point theory Leray-Schauder nonlinear alternative and lower and upper solutions method and so on, to investigate the existence of solutions of multi-point boundary value problem of nonlinear fractional differential equations.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 This paper investigates nonlinear fractional multi-point boundary value problem where n-1<α≤n,n-2<β0,0<ηi<1,(i= 1,2,…,m-2)withη=∑i=1m-2aiηiα-β-1<1 and Dβis the standard Riemann-Liouville fractional derivative,f:[0,1]×R×Râ†'R is continuous.By applying Schauder fixed point theorem,Leray-Schauder nonlinear alternative and Banach contraction princi-ple,this chapter gives(2.1.1)has a non-trivial solution,at least one non-trivial solution and the only three cases of non-trivial solution.Chapter 3 This paper investigates nonlinear fractional multi-point boundary value problem where 2<γ≤3,ai>0,0<ξi<1,(i=1,2,…,m-2)with b=∑i=1m-2aiξiγ-2<1 and Dγis the standard Riemann-Liouville fractional derivative,f:[0,1]×R+×R+â†'R+is continuous.By means of lower and upper solutions method and Schauder fixed point theorem,Some results on the existence of positive solutions of(3.1.1)are obtained.Chapter 4 This paper investigate nonlinear fractional multi-point boundary value problem where n-1<α≤n,n-2<β0,0<η<1, with aηα-β-1<1. Here f satisfies the Caratheodory conditions on[0,1]×(0,∞)×R (f∈Car([0,1]×(0,∞)×R),f is positive,f(t,x,y)is singular at x=0,and Dβis the standard Riemann-Liouville fractional derivative.By means of Krasnosel'skii fixed point theorem on a cone,the existence of positive solutions of(4.1.1)are obtained.